Holography of mass-deformed M2-branes

0.69 MB·

by  Sangmo Cheon

SNUTP10-008
Holography of mass-deformed M2-branes
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1 Sangmo Cheon, Hee-Cheol Kim and Seok Kim
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Department of Physics and Astronomy & Center for Theoretical Physics,

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Seoul National University, Seoul 151-747, Korea.

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p E-mails: [email protected], [email protected], [email protected]
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0 Abstract
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: We find and study the gravity duals of the supersymmetric vacua of N = 6 mass-
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i deformed Chern-Simons-matter theory for M2-branes. The classical solution extends that
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of Lin, Lunin and Maldacena by introducing a Z quotient and discrete torsions. The
r k
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gravity vacua perfectly map to the recently identified supersymmetric field theory vacua.
WecalculatethemassesofBPSchargedparticlesintheweaklycoupledfieldtheory,which
agree with the classical open membrane analysis when both calculations are reliable. We
also comment on how non-relativistic conformal symmetry is realized in our gravity duals
in a non-geometric way.
Contents
1 Introduction 1
2 The gravity soluions of mass-deformed M2-branes 3
2.1 Supersymmetric vacua of the field theory . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Gravity solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Discrete torsions and fractional M2-branes . . . . . . . . . . . . . . . . . . . . . 11
2.4 M2 charge, symmetry and map to the field theory vacua . . . . . . . . . . . . . 17
3 Examples 21
4 Charged particles and their gravity duals 25
4.1 Spectrum of charged particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Remarks on non-relativistic conformal symmetry . . . . . . . . . . . . . . . . . . 30
5 Discussions 31
A Supersymmetry of gravity solutions and M2-branes 33
B SU(2) tensor representation of the vacua 36
C Mass and supersymmetry of charged particles 37
1 Introduction
With recent advance in M2-brane physics from Chern-Simons-matter theories [1, 2], it became
clear that many essential aspects of M2-branes can be understood only when we have good
controls over strongly coupled quantum field theories. The strong coupling physics is crucial
in supersymmetry enhancement and appearance of M-theory states from monopole operators
[2, 3, 4, 5, 6, 7], the partition function and Wilson loops on S3 [8], determination of exact
U(1) R-symmetry in N = 2 theories [9], the N3/2 degrees of freedom [10, 11, 12], to list a
few examples. Analogous phenomena are either absent or turn out to be substantially simpler
in 4 dimensional Yang-Mills theories for D3-branes. Also, the physics often depends on the
Chern-Simons level k in a more nontrivial way than the Yang-Mills coupling constant.
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The roles of strong coupling dynamics turn out to be even more important for understand-
ing M2-brane systems with mass gap. The simplest M2-brane theory with a mass gap is the
mass-deformed N =6 Chern-Simons-matter theory [13]. This theory has many discrete super-
symmetric vacua, whose classical solutions are first found in [14] and refined in [15]. When
Chern-Simons level k is 1, the gravity duals of the supersymmetric vacua are found by Lin,
Lunin and Maldacena [16]. See also [17]. A puzzle was that the number of the gravity solutions
is much smaller than that of the classical field theory vacua found in [14]. This puzzle was
recently resolved in [15]. Many classically supersymmetric vacua of [14, 15] dynamically break
supersymmetry, afterwhichoneobtainsaperfectagreementbetweenthesupersymmetricvacua
of gauge theory and gravity at k=1.1
The analysis of [15] provides the partition function (or more precisely, the index) of super-
symmetric vacua at arbitrary Chern-Simons level k, whose gravity duals are not well under-
stood. The goal of this paper is to identify and study the gravity duals of mass-deformed N =6
theory at general coupling k.
Our results are very simple. The gravity duals for general k can be obtained from those for
k=1 [16] by introducing Z quotients, similar to the conformal Chern-Simons-matter theory
k
[2]. This fact has been already noticed and partially studied in [18]. Another important aspect
is that, contrary to the orbifold of AdS ×S7, the orbifold of the mass-deformed geometry has
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fixed points of the local form R8/Z . To correctly understand the gravity duals, one has to
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take into account the degrees of freedom localized on these fixed points. We find that fractional
M2-branes [19] are stuck to some of these fixed points, which appear in the gravity solutions
as discrete torsions. We find a set of gravity vacua which are in 1-to-1 map to the field theory
vacua of [15], showing correct properties to be the gravity duals of the latter.
Having obtained the precise map between the gravity backgrounds and the field theory
vacua, it could be possible to precisely address many questions on the gauge/gravity duality of
this system. For instance, the vortex solitons [20, 18] in this theory have been studied from the
gravity duals [18] using the probe D0-brane analysis at large k. The results of this paper may
help resolve some of the puzzles concerning these objects, raised in [18].
M2-brane systems with mass-gap could also have potential applications to low dimensional
condensed matter systems. For instance, it is well known that quantum Hall systems admit
a low energy description based on Chern-Simons theory. Addition of quasi-particles to this
system would yield a Chern-Simons theory coupled to massive charged matters. There are
some studies of (fractional) quantum Hall systems based on conformal Chern-Simons-matter
1It might sound surprising that the N=6 theory admits dynamical supersymmetry breaking. The analysis
of [15] is from the ‘UV completion’ of this theory (Yang-Mills Chern-Simons-matter theory) which can have
no more than N=3 supersymmetry. The result remains the same as one flows to IR, continuously taking the
Yang-Mills mass scale back to infinity.
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theories [21]. As the quantum Hall systems are gapped in the bulk, it should be interesting to
see if one can refine these studies with the mass-deformed M2-brane systems.
Havingsuchfuturedirections aside, westudy somebasicproperties ofvarious vacua, suchas
thespectrumofelementaryexcitations. AftersomepartofthegaugesymmetryU(N)×U(N)is
Higgsedinavacuum,theunbrokenpartofthegaugegroupsometimesexhibitsnon-perturbative
dynamics, similar to the confinement in N =1 Yang-Mills Chern-Simons theories studied in
[22, 23]. This fact can be captured by recently studied Seiberg-like dualities in 3 dimensions
[19], and also is correctly encoded in our gravity dual. We also study the gravity duals of
massive charged particles, given by open membranes connecting various fractional M2-branes
at the orbifold fixed points. The perturbative field theory analysis of BPS charged particles is
reliable when the ’t Hooft couplings of unbroken gauge groups are small, while classical open
M2-brane analysis is reliable when the membrane is macroscopic. We find a good agreement
between the two spectra when both calculations are reliable. Although we only discuss BPS
particles in this paper, similar comparison can be made with non-BPS particles.
The remaining part of this paper is organized as follows. In section 2, we first explain
the results of [15] which identifies the supersymmetric vacua of the field theory. Then after
explaining the gravity duals of the field theory vacua at k = 1, we introduce a Z orbifold
k
and discrete torsions for general Chern-Simons level k. Taking the fractional M2-branes into
account, we show that the supersymmetric ground states of the gravity solutions perfectly
map to the field theory vacua, with correct quantitative properties like symmetry or M2-brane
charges. The general considerations are illustrated by some examples in section 3. In section 4,
we study some elementary excitations in the context of gauge-gravity duality. A brief comment
on non-relativistic conformal symmetry is also given. In section 5, we conclude with discussions
on future directions. Appendix A summarizes the supersymmetry of gravity solutions and
various probe branes. Appendix B explains the SU(2) tensor description of the classical vacua.
Appendix C presents the mass calculation of charged particles from the field theory.
2 The gravity soluions of mass-deformed M2-branes
Chern-Simons-matter theory with mass deformation preserving N = 6 Poincare supersym-
metry was constructed and discussed in [13, 14]. The theory has four complex scalars in
bi-fundamental representations of the U(N) × U(N) gauge group, where the subscripts k
k −k
and −k denote the Chern-Simons levels. The classical and quantum supersymmetric vacua of
this theory have been studied in [14] and [15], respectively. At k=1, the gravity duals of the
quantum supersymmetric vacua are constructed in [16], which are asymptotic to AdS × S7
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in UV and exhibit complicated ‘bubbles’ of M2-branes polarized into M5-branes [24]. After
semi-classical quantization of the 4-form fluxes, the discretized gravity solutions are in 1-to-1
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correspondence to the supersymmetric vacua [15].
In this section, we review the supersymmetric vacua of the field theory identified in [15] for
general k, explain the gravity duals for k=1 obtained in [16], and present the generalization
for arbitrary k. Then we propose the map between the gravity solutions and the field theory
vacua with various evidences.
2.1 Supersymmetric vacua of the field theory
Before mass deformation, there is an SU(4) R-symmetry which rotates the four complex scalars
Z (for I = 1,2,3,4) in the fundamental representation. The mass deformation breaks this R-
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symmetry to SU(2) ×SU(2) ×U(1), where SU(2) and SU(2) rotate Z ,Z and Z ,Z as
1 2 1 2 1 2 3 4
doublets, respectively. See [13, 14, 15] for the details.
Theclassicalsupersymmetricvacuaaregivenbyscalarconfigurationswithvanishingbosonic
potential. The general solution is given by a direct sum of the following blocks. The blocks of
first type are n×(n+1) rectangular matrices with one more column, with n=0,1,2,···. The
blocks with n=0 denote empty columns. In this block, two scalars Z ,Z are taken to be zero,
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while the other two scalars are (µ is the mass parameter [15])
√
   
n 0 0 1
√ √
 n−1 0   0 2 
   
Z1 = µ12  ... ... , Z2 = µ12  ... ... . (2.1)
 √   √ 
   
2 0 0 n−1
   
√
1 0 0 n
Below, we shall call this the n’th block of first type. The blocks of second type are (n+1)×n
rectangular matrices with one more row, again with n=0,1,2,···. Blocks with n=0 denote
empty rows. Here the two scalars Z ,Z are zero, while
1 2
 √   
n 0
√
 0 n−1   1 0 
   
 0 ...   √2 ... 
Z3 = µ21  ... √2  , Z4 = µ12  ... 0 . (2.2)
   √ 
 0 1   n−1 0 
   
√
0 n
We shall call this the n’th block of second type. The classical supersymmetric vacua are
parametrized by specifying how many blocks of different types and sizes are included in the
direct sum. We denote by N the number of the n’th block of first type, and by N(cid:48) that of
n n
the n’th block of second type. An example of our parametrization is shown in Fig 1. As the
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N0=3 N1=2 N2=2
N0=3
Figure 1: An example of the parametrization of classical vacuum for N=11. Grey boxes denote
blocks with nonzero Z ,Z or Z ,Z . In this figure, the occupation numbers are N =3, N =2,
1 2 3 4 0 2
N(cid:48)=3, N(cid:48)=2, with all other numbers being zero.
0 1
direct sum (including N(cid:48) empty rows and N empty columns) should form an N ×N matrix,
0 0
we obtain the following constraints on these ‘occupation numbers’ {N ,N(cid:48)}:
n n
∞ ∞
(cid:88) (cid:88)
[nN +(n+1)N(cid:48)] = N , [(n+1)N +nN(cid:48)] = N . (2.3)
n n n n
n=0 n=0
The first and second conditions are restrictions on the numbers of rows and columns. Equiva-
lently, we obtain the following two constraints
∞ (cid:18) (cid:19) ∞ ∞
(cid:88) 1 (cid:88) (cid:88)
n+ (N +N(cid:48)) = N , N = N(cid:48) , (2.4)
2 n n n n
n=0 n=0 n=0
which will be a more suggestive form for later interpretation.
Among these classical zero energy solutions, only some of them remain to be exactly su-
persymmetric at the quantum level. We take the Chern-Simons level k to be positive without
losing generality. The vacua which survive to be supersymmetric quantum mechanically should
satisfy [15]
0 ≤ N ≤ k , 0 ≤ N(cid:48) ≤ k (2.5)
n n
for all occupation numbers. Furthermore, when these restrictions are satisfied, the degeneracy
(or more precisely the Witten index) of the supersymmetric vacua is
(cid:32) (cid:33)(cid:32) (cid:33)
∞
(cid:89) k k
, (2.6)
N N(cid:48)
n=1 n n
5
6
5
4
3
E=0
2 6
5
4
1 3
2
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Figure 2: A colored droplet for k=3 with (cid:96)=0. Droplets have Fermi energies E = 0,−2,2
F
from left to right. The Young diagrams with charges p=0,−2,2 correspond to the droplets:
the edge lengths of the diagrams and the droplets match, as illustrated by the numbers.
for given {N ,N(cid:48)} [15]. The total number of supersymmetric vacua with given rank N (i.e. for
n n
a given theory) is the summation of the degeneracy of the form (2.6), taken for all occupation
numbers satisfying (2.4) and (2.5).
One can also generalize the identification of supersymmetric vacua in [15] to the mass-
deformed N =6 theory with U(N) ×U(N+(cid:96)) gauge group, where 0 ≤ (cid:96) < k [19]. The only
k −k
change for the classical supersymmetric vacua is to replace (2.4) by
∞ (cid:18) (cid:19)
(cid:88) 1 (cid:96) (cid:88)
n+ (N +N(cid:48)) = N + , (N −N(cid:48)) = (cid:96) , (2.7)
2 n n 2 n n
n=0 n
while the conditions (2.5) and the degeneracy (2.6) remain the same.
Another way of viewing these supersymmetric vacua is to use k species of fermions, or
‘colored’ fermions, with occupation numbers N , N(cid:48) (for i = 1,2,··· ,k) being either 0 or 1.
ni ni
Being blind to the k ‘color’ quantum numbers, one obtains the combinatoric factor (2.6) in the
degeneracy by taking N =(cid:80)k N , N(cid:48) =(cid:80)k N(cid:48) . The first condition of (2.4) about the
n i=1 ni n i=1 ni
rank N is simply the overall energy condition for k pairs of chiral fermions in 1+1 dimensions.
As the energy level n+ 1 is half-integral, the fermions are in the Neveu-Schwarz sector. The
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second condition of (2.4) becomes the overall U(1) singlet condition for the sum over the U(1)k
color charges, where the particles with the occupation numbers N carry charge +1 and anti-
ni
particles with occupation numbers N(cid:48) carry charge −1. In other words, this condition sets the
ni
sum of k Fermi energy levels to be 0. For the theory with nonzero (cid:96), the last condition modifies
to setting the sum of Fermi levels to be (cid:96). Such fermion viewpoint of the occupations can be
illustrated as ‘droplets’ like Fig.2. In each droplet, an occupied sector with N =1 or N(cid:48) =1
ni ni
is represented by filling the n’th level above/beneath the Fermi level (denoted by E=0) with
a black/white stripe, respectively. In this example, the field theory occupation numbers are
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N(cid:48)=1, N(cid:48)=2, N(cid:48)=1, N(cid:48)=1, N =2, N =1, N =2, and others zero.
4 3 2 1 1 2 3
One can also bosonize the k colored fermions to k chiral bosons on a circle, namely to k
compact bosons. The U(1)k charges become the quantized momenta p (for i=1,2,··· ,k) of k
i
bosons on the circle. The neutrality of overall U(1) (or sum of Fermi levels being (cid:96)) corresponds
to the restriction
p +p +···+p = (cid:96) . (2.8)
1 2 k
The possible excitations of the k bosons map to the so-called colored partitions of N, which
consists of k Young diagrams with U(1)k charges {p }. The first condition of (2.4) on the energy
i
of k fermions can be rewritten in the bosonized picture as
k k ∞
(cid:96) 1 (cid:88) (cid:88)(cid:88)
N + = p2 + nN , (2.9)
2 2 i ni
i=1 i=1 n=1
where N = 0,1,2,··· are excitations of the bosonic oscillators (not to be confused with
ni
fermionic occupations with same notation above), and {p } are subject to the constraint (2.8).
i
Note that the first term on the right hand side of (2.9) is the kinetic energy of k zero modes.
In the bosonized picture, it is easy to calculate the partition function Z(q) for the super-
symmetric vacua, which trades the energy with the chemical potential q as Z(q) = Tr[qN+(cid:96)].
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The partition function for the k compact bosons with net momentum (cid:96) is
∞
Ik(q) = (cid:89) (1−1qn)k (cid:88) q12(cid:80)ki=1p2i . (2.10)
n=1 p1+p2+···+pk=(cid:96)
The first factor comes from the k Young diagrams, while the second factor is from the kinetic
energy of zero modes. This is a generalization of the result of [15] for (cid:96) = 0. For k = 1,
I (q)=(cid:81)∞ 1 agrees with the degeneracy of the gravity solutions of [16], as shown in [15].
1 n=1 1−qn
Strictly speaking, the above vacua are obtained by deforming the theory appropriately, under
which only the Witten index is invariant. So this partition function is the Witten index of the
field theory.
2.2 Gravity solutions
The gravity solutions for the supersymmetric mass-deformed M2-branes at Chern-Simons level
k=1, all asymptotic to AdS ×S7, are obtained in [16]. The metric and the 4-form field are
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S3
S3
x
Figure 3: The black/white regions with boundary conditions z(x,y)=∓1. The 4-sphere on the
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left combines a segment ending on black regions with S3 shrinking at the ends. The second
type of 4-sphere ends on white regions, and contains S˜3.
given by 2
(cid:104) (cid:105)
ds2 = e4Φ (cid:0)−dt2 +dw2 +dw2(cid:1)+e−2Φ h2(dy2 +dx2)+yeGdΩ2 +ye−GdΩ˜2
3 3
1 2 3 3
(cid:2) (cid:3)
e−2Φ = µ−2 h2 −h−2V2
0
G = −d(cid:0)e2Φh−2V(cid:1)∧dt∧dw ∧dw +µ−1(cid:2)Vd(y2e−2G)−h2e−3G (cid:63) d(y2e2G)(cid:3)∧dΩ˜
4 1 2 0 2 3
(cid:2) (cid:3)
+µ−1 Vd(y2e2G)+h2e3G (cid:63) d(y2e−2G) ∧dΩ , (2.11)
0 2 3
˜
where dΩ , dΩ denote length elements or volume 3-forms of unit round 3-spheres, which we
3 3
call S3, S˜3. Various functions in the solution are determined by two functions z(x,y), V(x,y),
2(cid:88)n+1 (−1)i+1(x−x ) 2(cid:88)n+1 (−1)i+1
i
z(x,y) = , V(x,y) = , ydV = −(cid:63) dz ((cid:15) = 1) .
(cid:112) (cid:112) 2 yx
2 (x−x )2 +y2 2 (x−x )2 +y2
i=1 i i=1 i
(2.12)
The functions G and h are given by z = 1 tanhG, h−2 = 2ycoshG. Note that, compared to
2
the solutions presented in [16], a parameter µ with dimension of mass is restored. This could
0
be eliminated to, say µ =1, by using the asymptotic conformal symmetry. µ will be identified
0 0
with the mass paremeter µ appearing in the field theory [15] as µ = πµ.
0 2k
From the metric in (2.11), y2 is proportional to the product of the square-radii of S3 and
S˜3. Therefore, at least one of the two 3-spheres shrink at y=0. For the geometry to be smooth
2The 4-form flux G corrects the expression in [16],
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[G ] =−d(cid:0)e2Φh−2V(cid:1)∧dt∧dw ∧dw − 1e−2Φ(cid:104)e−3G(cid:63) d(y2e2G)∧dΩ˜ +e3G(cid:63) d(y2e−2G)∧dΩ (cid:105) .
4 LLM 1 2 4 2 3 2 3
which we think should contain typos. In particular, we explicitly checked that the latter 4-form is not closed.
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with a shrinking 3-sphere, the 3-sphere should combine with the radial direction (∼ y) to form
R4. This requires the function z in (2.12) to have the boundary condition z(x,0) = ∓1, where
2
S3 or S˜3 shrinks for ∓ sign, respectively [16]. At the line parametrized by x at y = 0, we
therefore denote the parts with boundary behaviors z = ∓1 by black/white regions, as shown
2
in Fig 3. To visualize the regions better, we add a fictitious line segment to make the x line
look like an infinite strip of ‘droplet.’ In type IIB dual, this extra segment has the meaning
of a spatial direction called x− in [16], which is T-dualized to one of the spatial coordinates of
R2,1 in (2.11). To have asymptotic AdS ×S7, one should have a semi-infinite black region at
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one end and a white region at the other end. At the boundary of the adjacent black and white
regions (call it x=x for i = 1,2,···2n+1), both 3-spheres shrink and R8 appears near y=0,
i
x=x by combining the two 3-spheres with x,y.
i
There are various topological 4-cycles in this solution. Consider first a segment in the xy
plane ending on different black regions at y=0, and attach the 3-sphere S3 to it, like the cycle
on the left side of Fig 3. As S3 shrinks at the ends of the segment, the 4-cycle smoothly wraps
up, forming a 4-sphere. Similarly, one can consider a segment ending on different white regions
at y=0 and attach S˜3 to it, which also becomes a 4-sphere as shown by the cycle on the right
side of Fig 3. Nonzero 4-form fluxes are applied through these 4-spheres, which have to be
quantized. Below, we explain this quantization directly in M-theory. Similar discussion was
provided in [16] from the type IIB duals.
Consider a 4-sphere containing S˜3 which surrounds a black region (z=−1) between x=x
2 2j
and x , where j = 0,1,··· ,n: see the cycle on the right side of Fig 3. As the 4-form field is
2j+1
closed, we can deform the 4-sphere without changing the 4-form flux over the cycle. We take
the two points of the 4-sphere at y=0 to end exactly at the boundaries of the black region. We
also deform the whole 4-sphere to y=0. Near this black region, with small y, one obtains
(cid:34) (cid:35)
1 y2 (cid:88)2j (−1)i+1 2(cid:88)n+1 (−1)i+1 1 y2 (cid:88) (−1)i+1 1
˜
z(x,y) ≈ − + − + ≡ − + f(x), V(x,y) ≈ ≡ g˜(x).
2 4 (x−x )2 (x−x )2 2 4 2|x−x | 2
i i i
i=1 i=2j+1 i
(2.13)
Other functions are given by
y2 1 (cid:16) (cid:17)
e2G ≈ f˜(x) , h−2 ≈ ye−G ≈ 2f˜−1/2(x) , e−2Φ = h2 −h−2V2 ≈ f˜−g˜2 . (2.14)
4 2f˜1/2
Here, from
(cid:88)2j (−1)i+1 2(cid:88)n+1 (−1)i+1
g˜(x) = − , (2.15)
x−x x−x
i i
i=1 i=2j+1
one finds that
g˜(cid:48) = f˜. (2.16)
˜
One can easily check that f >0 for x <x<x , implying that g˜ is an increasing function
2j 2j+1
there. The case with j=0 (with x =−∞) can be regarded as the semi-infinite black region,
0
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